Question

What is fg(x) given that f(x) equals x² + 7x and g(x) equals -2x² + 3x-5

Answer 1

`\huge{ \boxed{f(g(x)) = 4 {x}^(4) - 12 {x}^(3) + 15 {x}^(2) - 9x - 10 \\ }}`

Explanation:

`f(x) = {x}^(2) + 7x \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ g(x) = - 2 {x}^(2) + 3x - 5`

To find f(g(x)), substitute g(x) into f(x). That is for every x found in f(x) replace it with g(x) and solve.

`f(g(x)) = {( - 2 {x}^(2) + 3x - 5) }^(2) + 7( - {2x}^(2) + 3x - 5) \: \: \: \: \: \: \: \: \: \: \\ \\ = (4 x^4 - 12 x^3 + 29 x^2 - 30 x + 25) + (-14 x^2 + 21 x - 35) \\ = 4 {x}^(4) - 12 {x}^(3) + 29 {x}^(2) - 14 {x}^(2) - 30x + 21x + 25 - 35 \\ \\ = 4 {x}^(4) - 12 {x}^(3) + 15 {x}^(2) - 9x - 10`

We have the final answer as

`f(g(x)) = 4 {x}^(4) - 12 {x}^(3) + 15 {x}^(2) - 9x - 10 \\`